Structure-preserving Lift & Learn: Scientific machine learning for nonlinear conservative partial differential equations
Boris Kramer
University of California San Diego
Mechanical and Aerospace Engineering
Conservative Hamiltonian systems are models commonly found in high-energy and plasma physics, quantum mechanics, and many engineering domains. These systems exhibit physically interpretable quantities such as momentum, energy, or vorticity; the behavior of these quantities in numerical simulation provides an important measure of accuracy of the model. Yet their simulation can also be expensive.
This talk will first give an overview of a few recently developed approaches for learning structure-preserving (low-dimensional) models and then introduce Hamiltonian Operator Inference, as well as Structure-preserving Lift & Learn, a scientific machine learning method that employs lifting variable transformations to learn structure-preserving reduced-order models for nonlinear partial differential equations (PDEs) with conservation laws. The work leverages a hybrid learning approach based on a recently developed energy-quadratization strategy that uses knowledge of the nonlinearity at the PDE level to derive an equivalent quadratic lifted system with quadratic system energy. Based on the lifting transformations and lifted model form, the proposed method derives quadratic reduced terms analytically and then uses those derived terms to formulate a constrained optimization problem to learn the remaining linear reduced operators in a structure-preserving way.
The proposed hybrid learning approach yields computationally efficient quadratic reduced-order models that respect the underlying physics of the high-dimensional problem. We demonstrate the generalizability of quadratic models learned via the proposed structure-preserving Lift & Learn method through three numerical examples: the one-dimensional wave equation with exponential nonlinearity, the two-dimensional sine-Gordon equation, and the two-dimensional Klein-Gordon-Zakharov equations, used in the study of the dynamics of Langmuir turbulence in plasmas.